Modulo Counting Research Articles

Modulo-counting first-order logic on bounded expansion classes

We prove that, on bounded expansion classes, every first-order formula with modulo counting is equivalent, in a linear-time computable monadic expansion, to an existential first-order formula.As a consequence, we derive, on bounded expansion classes, that first-order transductions with modulo counting have the same encoding power as existential first-order transductions. Also, modulo-counting first-order model checking and computation of the size of sets definable in modulo-counting first-order logic can be achieved in linear time on bounded expansion classes.As an application, we prove that a class has structurally bounded expansion if and only if it is a class of bounded depth vertex-minors of graphs in a bounded expansion class.We also show how our results can be used to implement fast matrix calculus on bounded expansion matrices over a finite field.

Long-distance phonological processes as tier-based strictly local functions

Whether we analyze phonological processes using a system of rules or constraints, the resulting map from underlying representations to surface pronunciations can be characterized as a function. Viewing processes as mathematical objects in this way allows us to study properties of phonology that hold no matter how it is implemented. Work in this vein has found that a majority of phonological processes only consider information within a finite window, placing them in the highly restrictive class of Strictly Local (SL) functions (Chandlee 2014; Chandlee et al. 2014; 2015). Long-distance phonological processes, however, lie outside the capabilities of the SL functions since they consider information that can be arbitrarily distant. The more powerful class of subsequential functions has been offered as a potential characterization of long-distance phonology (Heinz & Lai 2013; Luo 2017; Payne 2017), but we argue that an intermediate class offers a more natural model. Specifically, by incorporating an autosegmental tier (e.g., Goldsmith 1976) into the structure of an SL function, the non-local information crucial for applying long- distance processes can be rendered local. In addition to assessing the typological coverage of these Tier-based Strictly Local functions (Burness & McMullin 2019; Hao & Andersson 2019; Hao & Bowers 2019), we show that they fail to generate two pathological behaviours (minimum distance requirements and modulo counting) that can be accomplished with a subsequential function. We therefore conclude that tier-based computation is a better characterization of long-distance phonology than subsequential computation.

One-variable logic meets Presburger arithmetic

We consider the one-variable fragment of first-order logic extended with Presburger constraints. The logic is designed in such a way that it subsumes the previously-known fragments extended with counting, modulo counting or cardinality comparison and combines their expressive powers. We prove ▪-completeness of the logic by presenting an optimal algorithm for solving its finite satisfiability problem.

On the locality of arb-invariant first-order formulas with modulo counting quantifiers

We study Gaifman locality and Hanf locality of an extension of first-order logic with modulo p counting quantifiers (FO+MOD_p, for short) with arbitrary numerical predicates. We require that the validity of formulas is independent of the particular interpretation of the numerical predicates and refer to such formulas as arb-invariant formulas. This paper gives a detailed picture of locality and non-locality properties of arb-invariant FO+MOD_p. For example, on the class of all finite structures, for any p >= 2, arb-invariant FO+MOD_p is neither Hanf nor Gaifman local with respect to a sublinear locality radius. However, in case that p is an odd prime power, it is weakly Gaifman local with a polylogarithmic locality radius. And when restricting attention to the class of string structures, for odd prime powers p, arb-invariant FO+MOD_p is both Hanf and Gaifman local with a polylogarithmic locality radius. Our negative results build on examples of order-invariant FO+MOD_p formulas presented in Niemist\"o's PhD thesis. Our positive results make use of the close connection between FO+MOD_p and Boolean circuits built from NOT-gates and AND-, OR-, and MOD_p- gates of arbitrary fan-in.

Characterizing weighted MSO for trees by branching transitive closure logics

We introduce the branching transitive closure operator on progressing weighted monadic second-order logic formulas where the branching corresponds in a natural way to the branching inherent in trees. For arbitrary commutative semirings, we prove that weighted monadic second order logics on trees is equivalent to the definability by formulas which start with one of the following operators: (i) a branching transitive closure or (ii) one existential second-order quantifier followed by one universal first-order quantifier; in both cases the operator is applied to step-formulas over (a) Boolean first-order logic enriched by modulo counting or (b) Boolean monadic-second order logic.

Expressive Completeness for LTL With Modulo Counting and Group Quantifiers

Kamp showed that linear temporal logic is expressively complete for first order logic over words. We give a Gabbay style proof to show that linear temporal logic extended with modulo counting and group quantifiers (introduced by Baziramwabo, McKenzie, Thérien) is expressively complete for first order logic with modulo counting (introduced by Straubing, Thérien, Thomas) and group quantifiers (introduced by Barrington, Immerman, Straubing).